Nonuniform Steady States Research Articles

Emergence of non-trivial solutions from trivial solutions in reaction–diffusion equations for pattern formation

Reaction–diffusion equations serve as fundamental tools in describing pattern formation in biology. In these models, nonuniform steady states often represent stationary spatial patterns. Notably, these steady states are not unique, and unveiling them mathematically presents challenges. In this paper, we introduce a framework based on bifurcation theory to address pattern formation problems, specifically examining whether nonuniform steady states can bifurcate from trivial ones. Furthermore, we employ linear stability analysis to investigate the stability of the trivial steady-state solutions. We apply the method to two classic reaction–diffusion models: the Schnakenberg model and the Gray–Scott model. For both models, our approach effectively reveals many nonuniform steady states and assesses the stability of the trivial solution. Numerical computations are also presented to validate the solution structures for these models.

Coexistence of distinct nonuniform nonequilibrium steady states in Ehrenfest multiurn model on a ring.

The recently proposed Ehrenfest M-urn model with interactions on a ring is considered as a paradigm model which can exhibit a variety of distinct nonequilibrium steady states. Unlike the previous three-urn model on a ring which consists of a uniform steady state and a nonuniform nonequilibrium steady state, it is found that for even M≥4, an additional nonequilibrium steady state can coexist with the original ones. Detailed analysis reveals that this additional nonequilibrium steady state emerged via a pitchfork bifurcation which cannot occur if M is odd. Properties of this nonequilibrium steady state, such as stability, and steady-state flux are derived analytically for the four-urn case. The full phase diagram with the phase boundaries is also derived explicitly. The associated thermodynamic stability is also analyzed, confirming its stability. These theoretical results are also explicitly verified by direct Monte Carlo simulations for the three-urn and four-urn ring models.

Nonequilibrium steady states in coupled asymmetric and symmetric exclusion processes.

We propose and study a one-dimensional (1D) model consisting of two lanes with open boundaries. One of the lanes executes diffusive and the other lane driven unidirectional or asymmetric exclusion dynamics, which are mutually coupled through particle exchanges in the bulk. We elucidate the generic nonuniform steady states in this model. We show that in a parameter regime, where hopping along the TASEP lane, diffusion along the SEP lane, and the exchange of particles between the TASEP and SEP lanes compete, the SEP diffusivity D appears as a tuning parameter for both the SEP and TASEP densities for a given exchange rate in the nonequilibrium steady states of this model. Indeed, D can be tuned to achieve phase coexistence in the asymmetric exclusion dynamics together with spatially smoothly varying density in the diffusive dynamics in the steady state. We obtain phase diagrams of the model using mean field theories, and corroborate and complement the results with stochastic Monte Carlo simulations. This model reduces to an isolated open totally asymmetric exclusion process (TASEP) and an open TASEP with bulk particle nonconserving Langmuir kinetics (LK), respectively, in the limits of vanishing and diverging particle diffusivity in the lane executing diffusive dynamics. Thus, this model works as an overarching general model, connecting both pure TASEPs and TASEPs with LK in different asymptotic limits. We further define phases in the SEP and obtain phase diagrams and show their correspondence with the TASEP phases. In addition to its significance as a 1D driven, diffusive model, this model also serves as a simple reduced model for cell biological transport by molecular motors undergoing diffusive and directed motion inside eukaryotic cells.

On a hypercycle equation with infinitely many members

A hypercycle equation with infinitely many types of macromolecules is formulated and studied both analytically and numerically. The resulting model is given by an integro-differential equation of the mixed type. Sufficient conditions for the existence, uniqueness, and non-negativity of solutions are formulated and proved. Analytical evidence is provided for the existence of non-uniform (with respect to the second variable) steady states. Finally, numerical simulations strongly indicate the existence of a stable nonlinear wave in the form of the wave train.

Stabilization of the Linearized Water Tank System

In this article we study the so-called water tank system. In this system, the behavior of water contained in a 1-D tank is modelled by Saint-Venant equations, with a scalar distributed control. It is well-known that the linearized systems around uniform steady-states are not controllable, the uncontrollable part being of infinite dimension. Here we will focus on the linearized systems around non-uniform steady states, corresponding to a constant acceleration of the tank. We prove that these systems are controllable in Sobolev spaces, using the moments method and perturbative spectral estimates. Then, for steady states corresponding to small enough accelerations, we design an explicit Proportional Integral feedback law (obtained thanks to a well-chosen dynamic extension of the system) that stabilizes these systems exponentially with arbitrarily large decay rate. Our design relies on feedback equivalence/backstepping.

Spatial pattern formation in reaction-diffusion models: a computational approach.

Reaction-diffusion equations have been widely used to describe biological pattern formation. Nonuniform steady states of reaction-diffusion models correspond to stationary spatial patterns supported by these models. Frequently these steady states are not unique and correspond to various spatial patterns observed in biology. Traditionally, time-marching methods or steady state solvers based on Newton's method were used to compute such solutions. However, the solutions that these methods converge to highly depend on the initial conditions or guesses. In this paper, we present a systematic method to compute multiple nonuniform steady states for reaction-diffusion models and determine their dependence on model parameters. The method is based on homotopy continuation techniques and involves mesh refinement, which significantly reduces computational cost. The method generates one-parameter steady state bifurcation diagrams that may contain multiple unconnected components, as well as two-parameter solution maps that divide the parameter space into different regions according to the number of steady states. We applied the method to two classic reaction-diffusion models and compared our results with available theoretical analysis in the literature. The first is the Schnakenberg model which has been used to describe biological pattern formation due to diffusion-driven instability. The second is the Gray-Scott model which was proposed in the 1980s to describe autocatalytic glycolysis reactions. In each case, the method uncovers many, if not all, nonuniform steady states and their stabilities.

A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope

The exponential stability problem of the nonlinear Saint-Venant equations is addressed in this paper. We consider the general case where an arbitrary friction and space-varying slope are both included in the system, which lead to non-uniform steady-states. An explicit quadratic Lyapunov function as a weighted function of a small perturbation of the steady-states is constructed. Then we show that by a suitable choice of boundary feedback controls, that we give explicitly, the local exponential stability of the nonlinear Saint-Venant equations for the H2-norm is guaranteed.

Instability and dynamics of volatile thin films

Volatile viscous fluids on partially-wetting solid substrates can exhibit interesting interfacial instabilities and pattern formation. We study the dynamics of vapor condensation and fluid evaporation governed by a one-sided model in a low Reynolds number lubrication approximation incorporating surface tension, intermolecular effects and evaporative fluxes. Parameter ranges for evaporation- dominated and condensation-dominated regimes and a critical case are identified. Interfacial instabilities driven by the competition between the disjoining pressure and evaporative effects are studied via linear stability analysis. Transient pattern formation in nearly-flat evolving films in the critical case is investigated. In the weak evaporation limit unstable modes of finite amplitude non-uniform steady states lead to rich droplet dynamics, including flattening, symmetry breaking, and droplet merging. Numerical simulations show long time behaviors leading to evaporation or condensation are sensitive to transitions between film-wise and drop-wise dynamics.

A quadratic Lyapunov function for hyperbolic density–velocity systems with nonuniform steady states

A new explicit Lyapunov function allows to study the exponential stability for a class of physical ‘2 by 2’ hyperbolic systems with nonuniform steady states. In fluid dynamics, this class of systems involves isentropic Euler equations and Saint-Venant equations. The proposed quadratic Lyapunov function allows to analyze the local exponential stability of the system equilibria for suitable dissipative Dirichlet boundary conditions without additional conditions on the system parameters.

Two-layer electrified pressure-driven flow in topographically structured channels

The flow of two stratified viscous immiscible perfect dielectric fluids in a channel with topographically structured walls is investigated. The flow is driven by a streamwise pressure gradient and an electric field across the channel gap. This problem is explored in detail by deriving and studying a nonlinear evolution equation for the interface valid for large-amplitude long waves in the Stokes flow regime. For flat walls, the electrified flow is long-wave unstable with a critical cutoff wavenumber that increases linearly with the magnitude of the applied voltage. In the nonlinear regime, it is found that the presence of pressure-driven flow prevents electrostatically induced interface touchdown that has been observed previously – time-modulated nonlinear travelling waves emerge instead. When topography is present, linearly stable uniform flows become non-uniform spatially periodic steady states; a small-amplitude asymptotic theory is carried out and compared with computations. In the linearly unstable regime, intricate nonlinear structures emerge that depend, among other things, on the magnitude of the wall corrugations. For a low-amplitude sinusoidal boundary, time-modulated travelling waves are observed that are similar to those found for flat walls but are influenced by the geometry of the wall and slide over it without touching. The flow over a high-amplitude sinusoidal pattern is also examined in detail and it is found that for sufficiently large voltages the interface evolves to large-amplitude waves that span the channel and are subharmonic relative to the wall. A type of ‘walking’ motion emerges that causes the lower fluid to wash through the troughs and create strong vortices over the peaks of the lower boundary. Non-uniform steady states induced by the topography are calculated numerically for moderate and large values of the flow rate, and their stability is analysed using Floquet theory. The effect of large flow rates is also considered asymptotically to find solutions that compare very well with numerical computations.

Falling liquid films with blowing and suction

Flow of a thin viscous film down a flat inclined plane becomes unstable to long-wave interfacial fluctuations when the Reynolds number based on the mean film thickness becomes larger than a critical value (this value decreases as the angle of inclination to the horizontal increases, and in particular becomes zero when the plate is vertical). Control of these interfacial instabilities is relevant to a wide range of industrial applications including coating processes and heat or mass transfer systems. This study considers the effect of blowing and suction through the substrate in order to construct from first principles physically realistic models that can be used for detailed passive and active control studies of direct relevance to possible experiments. Two different long-wave, thin-film equations are derived to describe this system; these include the imposed blowing/suction as well as inertia, surface tension, gravity and viscosity. The case of spatially periodic blowing and suction is considered in detail and the bifurcation structure of forced steady states is explored numerically to predict that steady states cease to exist for sufficiently large suction speeds since the film locally thins to zero thickness, giving way to dry patches on the substrate. The linear stability of the resulting non-uniform steady states is investigated for perturbations of arbitrary wavelength, and any instabilities are followed into the fully nonlinear regime using time-dependent computations. The case of small amplitude blowing/suction is studied analytically both for steady states and their stability. Finally, the transition between travelling waves and non-uniform steady states is explored as the amplitude of blowing and suction is increased.

On boundary feedback stabilization of non-uniform linear [formula omitted] hyperbolic systems over a bounded interval

Conditions for boundary feedback stabilizability of non-uniform linear 2 × 2 hyperbolic systems over a bounded interval are investigated. The main result is to show that the existence of a basic quadratic control Lyapunov function requires that the solution of an associated ODE is defined on the considered interval. This result is used to give explicit conditions for the existence of stabilizing linear boundary feedback control laws. The analysis is illustrated with an application to the boundary feedback stabilization of open channels represented by linearized Saint–Venant equations with non-uniform steady-states.

Further results on boundary feedback stabilisation of 2 × 2 hyperbolic systems over a bounded interval

Conditions for boundary feedback stabilisability of linear 2×2 hyperbolic systems over a bounded interval are investigated. The main result is to show that the existence of a quadratic control Lyapunov function requires that the solution of an associated ODE is defined on the considered interval. This result is used to give explicit conditions for the existence of stabilising linear boundary feedback control laws. The analysis is illustrated with an application to the boundary feedback stabilisation of open channels represented by Saint-Venant equations with non-uniform steady-states. Copyright © IFAC 2010

The impact of bubble diffusivity on confined oscillated bubbly liquid

We consider the dynamics of monodisperse bubbly liquid confined by two plane solid walls and subject to small-amplitude high-frequency transverse oscillations. The period of these oscillations is assumed small in comparison with typical relaxation times for a single bubble but comparable with the period of volume eigenoscillations. The time-averaged description accounting for the two-way coupling between the liquid and the bubbles and for the diffusivity of bubbles is applied. We find nonuniform steady states with the liquid quiescent on average. At relatively low frequencies, accumulation of bubbles either at the walls or in planes parallel to the walls is detected. These one-dimensional states are shown to be unstable. At relatively high frequencies, this accumulation is found at the central plane and the solution is stable.

On a model of spatial spread of epidemics with long-distance travel

This Letter studies the dynamics of infectious diseases which are spread geographically through long-distance travel between two regions and subsequent local redistribution through a process of diffusion. A particular case of an equiproportionate travel is considered, and the model describes migration rather than short visits. We examine uniform and nonuniform steady states together with their linear stability. Numerical simulations are performed to illustrate the evolution of initial distribution of population towards its final stage, which is represented by uniform distribution of the total population among infected individuals.

Network topology and Turing instabilities in small arrays of diffusively coupled reactors

We study the effect of the network structure on the diffusion-induced instability to nonuniform steady states in arrays of diffusively coupled reactors. The kinetics is given by the Lengyel–Epstein model, and we derive the conditions for Turing instabilities in all arrays of two, three, and four reactors.

Pattern formation in the Gray–Scott model

We investigate an elliptic system that arises in cubic autocatalytic reactions known as the Gray–Scott model. Complicated patterns were reported by Pearson in a numerical study of this system. We produce the bifurcation analysis to support the existing numerical evidence for patterns. Specifically bifurcation results and C 2 bounds for nonuniform steady states are derived.

Traveling Waves and Nonuniform Steady States in Boiling

Traveling Waves and Nonuniform Steady States in Boiling

Comptetition in a Spatially Heterogeneous Environment: Modelling the Risk of Spread of a Genetically Engineered Population

In recent years regulations have been developed to address the risks of releasing genetically engineered organisms into the natural environment. These risks are generally considered to be proportional to the exposure multiplied by the hazard. Exposure is, in part, determined by the spatial spread of the organisms, a component of risk suited to mathematical analysis. In this paper we exampine a mathematical model describing the spread of organisms introduced into a hetereogeneous environment, focusing on the risk of spread and plausibility of containment strategies. Two competing populations are assumed, one the natural species and the other an engineered species or strain, both of which move randomly in a spatially heterogenous environment consisting of alternating favourable and unfavourable patches. The classical Lotka–Volterra competition model with diffusion is used. Analyses of the possible spread and invasion of engineered organisms are thus reduced to finding periodic travelling wave solutions to the model equations. We focus on whether a very small number of engineerd organisms can spatially invade a natural population. Initially we investigate the problem for spatially periodic diffusion coefficients and demonstrate that, under the right circumstances and a large enough unfavourable patch, invasion does not succeed. However, if spatially periodic carrying capacities are assumed along with spatially varying diffusion rates, the situation is far more complex. In this case containment of the engineered species is no longer only a simple function of the unfavourable patch length. By using perturbation solutions to the nonuniform steady states, approximate invasion conditions are obtained.

The effects of coupling on pattern formation in a simple autocatalytic system

The spatiotemporal structures that can arise in two identical cells, each governed by cubic autocatalator kinetics and coupled via the diffusive interchange of a reactant, are discussed. The coupling gives rise to five spatially uniform steady states, one of which exists in the uncoupled system. By studying the linearized equations, it is found that three of these steady states, including that of the uncoupled system, may give rise to the possibility of bifurcations to spatially nonuniform steady states. In the case of the steady state corresponding to that of the uncoupled system, it is seen that the coupling leads to bifurcations not present in the uncoupled system which give rise to locally stable nonuniform steady states. A weakly nonlinear analysis is developed for both small and large coupling strength a, and for parameter values in a neighbourhood of the bifurcation points on the new steady states. This clarifies the nature of the nonuniform solutions close to bifurcation, which are then followed numerically using a path-following technique. The coupling is found to produce extra nonuniform steady solutions which are stable close to their bifurcation points.